Asymptotic relations of the Bourgain-Brezis-Mironescu type for mappings between singular spaces

TL;DR

This paper extends Bourgain-Brezis-Mironescu type results to mappings between singular metric measure spaces, characterizing Sobolev and bounded variation maps through asymptotic analysis of nonlocal functionals.

Contribution

It provides the first characterization of Sobolev and BV maps via these nonlocal functionals in the singular setting, and derives explicit limit formulas for Sobolev maps.

Findings

Characterization of Sobolev maps via nonlocal functionals

Explicit limit formulas for Sobolev maps

Extension of Bourgain-Brezis-Mironescu results to singular spaces

Abstract

We explore the asymptotic behavior of families of Bourgain-Brezis-Mironescu type nonlocal functionals for mappings from metric measure spaces to arbitrary metric spaces. As the first outcome, we obtain a characterization of Sobolev maps and of maps of bounded variation via such functionals. As the second outcome, we establish precise expressions of the limits of such functionals for Sobolev maps. All this provides an extension of several Bourgain-Brezis-Mironescu type results to the entirely singular setting.